Question

law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$
in $\triangle bcd$, $d = 3$, $b = 5$, and $mangle d=25^{circ}$. what are the possible approximate measures of angle b?
only $90^{circ}$
only $155^{circ}$
$20^{circ}$ and $110^{circ}$
$45^{circ}$ and $135^{circ}$

Explanation:

Step1: Apply the law of sines

According to the law of sines $\frac{\sin(D)}{d}=\frac{\sin(B)}{b}$. Substitute $d = 3$, $b = 5$, and $m\angle D=25^{\circ}$ into the formula: $\frac{\sin(25^{\circ})}{3}=\frac{\sin(B)}{5}$.

Step2: Solve for $\sin(B)$

Cross - multiply to get $\sin(B)=\frac{5\sin(25^{\circ})}{3}$. Calculate $\sin(25^{\circ})\approx0.4226$, then $\sin(B)=\frac{5\times0.4226}{3}\approx0.7043$.

Step3: Find the possible values of angle $B$

Since $\sin(B)\approx0.7043$, $B=\sin^{- 1}(0.7043)\approx45^{\circ}$ or $B = 180^{\circ}-45^{\circ}=135^{\circ}$ (because $\sin\theta=\sin(180^{\circ}-\theta)$ for any angle $\theta$ in the range $0^{\circ}<\theta<180^{\circ}$).

Answer:

D. $45^{\circ}$ and $135^{\circ}$